Convex Functions and Raina’s Fractional Double Integrals

A convex analysis is the branch of mathematics in which we study the properties of convex sets and convex functions.ese classical concepts have a wide range of applications in both pure and applied sciences. For instance, every one is familiar with the role of convexity in the theory of optimization, operations research, mathematical economics, theory of means etc. In recent years the classical concepts of the convexity have been extended and generalized in dierent directions using novel and innovative ideas. For example, Dragomir [1] extended the notion of classical convex functions on the coordinates and introduced the class of co-ordinated convex functions. Iscan [2] introduced the notion of harmonically convex functions and observed that this class enjoys some nice properties which the convex functions have. Nikodem [3] introduced the class of interval-valued convex functions and discussed its properties. Zhao et al. [4] introduced the notion of interval-valued harmonically convex functions. For more details, interested readers are referred to the book [5]. Another charming aspect of the theory of convexity is its relation with the theory of inequalities. Many inequalities which are known to us are direct consequences of the applications of the convexity property of the functions. In this regard, one of the most studied results is Hermite-Hadamard’s inequality. is inequality provides us with a necessary and sucient condition for a function to be convex. It reads as, letΘ: I [℘1,℘2]⊆R⟶ R be a convex function, then


Introduction
A convex analysis is the branch of mathematics in which we study the properties of convex sets and convex functions. ese classical concepts have a wide range of applications in both pure and applied sciences. For instance, every one is familiar with the role of convexity in the theory of optimization, operations research, mathematical economics, theory of means etc. In recent years the classical concepts of the convexity have been extended and generalized in di erent directions using novel and innovative ideas. For example, Dragomir [1] extended the notion of classical convex functions on the coordinates and introduced the class of co-ordinated convex functions. Iscan [2] introduced the notion of harmonically convex functions and observed that this class enjoys some nice properties which the convex functions have. Nikodem [3] introduced the class of interval-valued convex functions and discussed its properties. Zhao et al. [4] introduced the notion of interval-valued harmonically convex functions. For more details, interested readers are referred to the book [5].
Another charming aspect of the theory of convexity is its relation with the theory of inequalities. Many inequalities which are known to us are direct consequences of the applications of the convexity property of the functions. In this regard, one of the most studied results is Hermite-Hadamard's inequality. is inequality provides us with a necessary and su cient condition for a function to be convex. It reads as, let Θ: I [℘ 1 , ℘ 2 ]⊆R ⟶ R be a convex function, then is result is one of the most signi cant results pertaining to the convexity property of the functions which has been studied extensively as well as intensively. In recent years this result has been extended and generalized in di erent ways using novel and innovative ideas. For example, Dragomir [1] obtained a new version of Hermite-Hdamard's inequality by using the co-ordinated convexity property of the functions. Iscan [2] obtained Hermite-Hadamard's inequality using the class of harmonic convex functions. Zhao obtained a similar result by using the interval-valued harmonically convex functions. Sarikaya et al. [6] have utilized the concepts of fractional calculus and obtained fractional analogues of Hermite-Hadamard's inequality. For some more recent studies regarding Hermite-Hadamard's inequality and its applications, see [7].
On the other hand, the interval analysis, which is used in mathematics and computer models as one of the ways for resolving interval uncertainty, is an important material in mathematics. Despite the fact that this theory has a lengthy history dating back to Archimedes' estimate of the circumference of a circle, a substantial research in this topic was not published until the 1950s. In 1966, Moore, the pioneer of interval calculus, released the first book [8] on interval analysis. Following that, a slew of researchers delved into the theory and applications of interval analysis. Many authors have recently focused on integral inequalities derived from interval-valued functions. Sadowska [9] discovered the Hermite-Hadamard inequality for set-valued functions, which is a more general form of interval-valued mappings: Theorem 1 (see [9]). Suppose that Θ: [℘ 1 , ℘ 2 ] ⟶ R + I is interval-valued convex function such that F(t) � [Θ(t), F(Θ)]. en, we have the following inequalities: Many papers have been devoted to generalizations of the inclusions (2). For example; Budak et al. [10] proved the Hermite-Hadamard inclusion by using Riemann-Liouville fractional integrals of interval-valued functions. Some papers focused on to generalization of (2) by utilizing some kinds of general convexities [4,[11][12][13]. Many authors also proved the corresponding Hermite-Hadamard inclusions for intervalvalued functions with two variables [14][15][16][17]. For interested readers, we refer to the following articles [18][19][20]. e main objective of this paper is to derive new fractional refinements of Hermite-Hadamard's inclusion using the class of intervalvalued co-ordinated harmonically convex functions.

Preliminaries
In this section, we discuss some preliminary concepts and results.
A set C⊆R is said to be convex, if Dragomir extended the notion of classical convexity and introduced the class of co-ordinated convex functions as follows.
Definition 2 (see [2]). A function Θ: Proposition 1 (see [2]) Theorem 3 (see [2]). Let Θ: Theorem 4 (see [21]). Suppose that Θ: Δ ⟶ R is coordinated harmonically convex function, then We now discuss some preliminaries from interval analysis. The idea behind the interval analysis is the observation that if you compute a number μ and a bound M on the total error in μ, as an approximation to some unknown number X, such that |X − μ| ≤ M, then no matter how you computed μ and M, you will know that X lies in the interval [μ − M, μ + M].
Let K ℘ 3 be a collection of all closed and bounded intervals of the form [w * , w * ] for all w * , w * ∈ R. If w * , w * ≥ 0, then [w * , w * ] is called a positive interval. Furthermore, K + ℘ 3 � [w * , w * ], ∀w * ≥ 0 is known as the set of all positive intervals.
We now discuss about interval arithmetics and related properties Unlike the usual arithmetics the interval addition, subtraction product and division is defined as follows: where Also, if w is an interval without zero element, then so quotient of u and w is defined as follows: Note that, addition, subtraction, product, and quotient of intervals is again an interval.
Let R I , R I + , and R I − are set of closed intervals, set of positive closed intervals and set of negative closed intervals, respectively, then following are some algebraic properties of intervals: All of these properties also hold for multiplication. But distributive law does not hold for intervals. e following example illustrates this fact.
For A 1 � [u * , u * ], A 2 � [w * , w * ] ∈ K ℘ 3 , the inclusion ⊆ is given by the following equation: Now, we rewrite the subsequent distance of intervals A 1 and A 2 is regarded as Hausdorff-Pompeiu distance as follows: We now discuss the integration of interval-valued functions. If be the set of all points P such that mesh(P) < ρ, then Θ: , if there exist K ∈ R I and for each ϵ > 0 there exists ρ > 0 such that where S(Θ, P, ρ) is the Riemann sum of Θ with respect to P ∈ B(ρ, [℘ 1 , ℘ 2 ]). (20) shows that K is the (IR)-integral of Θ and given by the following equation: Now, we discuss interval-valued integration of double integrals and generalized Raina's interval-valued integrals.

Theorem 6. Assume that the interval-valued function
For more details, see [8,22]. The class of interval-valued convex functions is defined as follows.
Definition 4 (see [3]). A function Θ: [℘ 1 , ℘ 2 ] ⟶ R + I is said to be interval-valued convex function, if the following inclusion holds: Now, we recall the definition of interval-valued harmonically convexity, which is defined as follows.

be interval-valued convex function if and if only Θ * (x) is a convex function and Θ * (x) is a concave function.
We now recall some concepts from fractional calculus. The classical Riemann-Liouville fractional integrals are defined as: Θ of order α > 0 are defined by the following equation: Sarikaya et al. [6] have obtained the following fractional analogue of Hermite-Hadamard's inequality: This motivated Iscan and Wu [24] and as result, they have obtained a fractional analogue of Hermite-Hadamard's inequality essentially using the class of harmonically convex functions.
Definition 9. Let Θ⊆R ⟶ R I be an interval-valued function and F be a Reimann interval-valued integrable function satisfying the condition Using the definitions of κ-Riemann-Liouville fractional integrals Noor et al. [21] have obtained κ-fractional analogue of Hermite-Hadamard's inequality.
We now introduce the κ-Raina's fractional double integrals, which are defined as follows. Here, s function which is defined as follows: where ρ, λ > 0, with bounded modulus |z| < M, and } is a bounded sequence of positive real numbers. For details, see [27].
In the following example, we give the numerical verification of Definition (10).
and Θ(t, s) � [st, s + t], then we have the following equation: Journal of Mathematics en, (34) can be written as follows: A combination of (35), (36), and (37) yields the required result.
e main objective of this article is to obtain some new integral inclusions essentially using the interval-valued harmonically co-ordinated convex functions and k-Raina's fractional double integrals. To show the validity of our theoretical results, we also give some numerical examples. We hope that the ideas and the techniques of this paper will inspire interested readers.

Results and Discussions
In this section, we discuss our main results. First of all, we introduce the class of interval-valued harmonically coordinated convex functions.
Proof. Using the definition of co-ordinated harmonic convex functions the proof is straight forward.

Journal of Mathematics
If Θ(x, y) is harmonically interval-valued concave function then we have the following equation: Proof. e proof is left for interested readers.
which gives the verification of eorem 9.
If Θ is interval-valued harmonically co-ordinated concave then the following relation holds: Proof. e proof is obvious by definition and some computation as well. where Proof 4. From the given hypothesis and interval-valued harmonically convexity property, we have the following equation: Adding (49), (50) and multiplying inequality by t (α/k)− 1 R σ,k ρ,w,α [w(℘ 1 ℘ 2 /℘ 2 − ℘ 1 ) ρ t ρ ]p, we obtain the following equation: Similarly, we can calculate, using interval-valued harmonically convexity, 8 Journal of Mathematics Combination of (51) and (52) refers to relation (47). is completes the proof.
Proof. e proof is left for interested readers. Firstly, we establish fractional Hermite-Hadamard inequality.
If Θ(x, y) is an interval-valued harmonically co-ordinated concave function then we have the following equation: Proof. From the definition of a harmonically co-ordinated convex function and according to a hypothesis, we have the following equation: Multiplying both sides of (56) by and integration with respect to (t, s) in [0, 1] × [0, 1], then we have the following equation: After simple computation, we have the following equation: From (57), we have the following equation: Combination of (71) and (60), we obtain the first inclusion of (54).
To prove our second inclusion, we use a definition of harmonically co-ordinated convexity.
Combining (66) and (67), we obtain the second part of our required result. is completes the proof. Now, we develop the new midpoint Hermite-Hadamard's inequality in the sense of interval-valued harmonically co-ordinated convexity.
, then the following inclusion holds: If Θ(x, y) is interval-valued harmonically co-ordinated concave function, then we have the following equation:

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Proof. From the definition of a harmonically co-ordinated convex function, we have the following equation: Multiplying both sides of (70) by and integration with respect to (t, s) in [0, 1] × [0, 1], then we have the following equation: After simple computation, we have the following equation: Similarly, we have the following equation: By the combination of (72) and (73), we obtain first inclusion of (68).
To prove our second inclusion, we use a definition of harmonically co-ordinated convexity.